CAT 2023 Logarithms Question (Hard)

Question

If and are positive real numbers such that and , then equals

Accepted Answer

Step 1 We are given log_ x x 2 12 4 By the definition of logarithm x 4 x 2 12 Move all terms to one side x 4 - x 2 - 12 0 Let t 2 - t - 12 0 Solve quadratic in t 2 2 Thus or Since , so . Thus as Step 2 We are also given 3 log_ y x 1 Rightarrow log_ y x 3 Therefore y 1 3 x Rightarrow y x 3 Since , St

CAT 2023

Quantitative

Logarithms

Hard

If $x$ and $y$ are positive real numbers such that $\log_{x}\left(x^2+12\right)=4$ and $3 \log_{y} x=1$ , then $x+y$ equals

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